Question

There is a pond near a grass garden. From a herd of deers, half of the deers are grazing in the garden. The half of the remaining number of deers are playing with one another. Further, the half part of the remaining deers are taking rest in the garden and 9 deers are drinking from the pond. How many deers will be in the group?

Answer 1

Hint: Here, we have to find the number of deers in the group. We will assume the number of deers in the group to be \[x\]. We will use the given information to form a linear equation in terms of \[x\]. Then, we will solve this equation to get the value of \[x\], and hence, the number of deers in the group.

Complete step-by-step answer:
Let the number of deers in the group be \[x\].
We will apply the stated operations on \[x\] to form an equation in terms of \[x\].
First, we know that half of the deers are grazing in the garden.
Therefore, we get
Number of deers grazing in the garden \[ = \dfrac{1}{2}{\rm{of }}x = \dfrac{1}{2}x = \dfrac{x}{2}\]
The number of remaining deers is the difference in the total number of deers in the group, and the number of deers grazing in the garden.
Therefore, we get
Number of remaining deers (not grazing in the garden) \[ = x - \dfrac{x}{2}\]
Taking the L.C.M. and subtracting, we get
Number of remaining deers (not grazing in the garden) \[ = \dfrac{{2x - x}}{2} = \dfrac{x}{2}\]
Now, half of the remaining deers are playing with one another.
Therefore, we get
Number of deers playing with one another \[ = \dfrac{1}{2}{\rm{of }}\dfrac{x}{2} = \dfrac{1}{2} \times \dfrac{x}{2}\]
Multiplying the terms of the expression, we get
Number of deers playing with one another \[ = \dfrac{x}{4}\]
The number of remaining deers is the difference in the number of deers remaining (not grazing in the garden), and the number of deers playing with one another.
Therefore, we get
Number of remaining deers (Not grazing or playing) \[ = \dfrac{x}{2} - \dfrac{x}{4}\]
Taking the L.C.M. and subtracting, we get
Number of remaining deers (not grazing or playing) \[ = \dfrac{{2x - x}}{4} = \dfrac{x}{4}\]
Next, half of the remaining deers are taking rest in the garden.
Therefore, we get
Number of deers taking rest in the garden \[ = \dfrac{1}{2}{\rm{of }}\dfrac{x}{4} = \dfrac{1}{2} \times \dfrac{x}{4}\]
Multiplying the terms of the expression, we get
Number of deers taking rest in the garden \[ = \dfrac{x}{8}\]
The number of remaining deers is the difference in the number of deers remaining (not grazing or playing), and the number of deers taking rest in the garden.
Therefore, we get
Number of remaining deers (Not grazing, playing, or resting) \[ = \dfrac{x}{4} - \dfrac{x}{8}\]
Taking the L.C.M. and subtracting, we get
Number of remaining deers (Not grazing, playing, or resting) \[ = \dfrac{{2x - x}}{8} = \dfrac{x}{8}\]
Now, the remaining deers are drinking water from the pond.
The number of remaining deers drinking water from the pond is the number of deers not grazing, not playing, and not resting.
Therefore, we get
Number of deers drinking water from the pond \[ = \dfrac{x}{8}\]
Finally, it is given that the number of deers drinking water from the pond is 9.
Therefore, we get
\[ \Rightarrow \dfrac{x}{8} = 9\]
Multiplying both sides by 8, we get
\[\begin{array}{l} \Rightarrow \dfrac{x}{8} \times 8 = 9 \times 8\\ \Rightarrow x = 72\end{array}\]
\[\therefore\] The total number of deers in the group is 72.

Note: We can verify our answer by using the information given in the question.
The total number of deers is 72.
Half of the total number of deers is grazing in the garden.
Therefore, we get
Number of deers grazing in the garden \[ = \dfrac{{72}}{2} = 36\]
Remaining number of deers \[ = 72 - 36 = 36\]
Now, half of the remaining number of deers are playing with one another.
Therefore, we get
Number of deers playing with one another \[ = \dfrac{{36}}{2} = 18\]
Remaining number of deers \[ = 36 - 18 = 18\]
Next, half of the remaining number of deers are taking rest in the garden.
Therefore, we get
Number of deers taking rest in the garden \[ = \dfrac{{18}}{2} = 9\]
Remaining number of deers \[ = 18 - 9 = 9\]
Therefore, the number of remaining deers is 9. This is the number of deers drinking water from the pond.
Hence, we have verified our answer.